Zeros of solutions of linear equations
Nontrivial (i.e., not identically zero) solutions of linear ordinary differential equations obviously possess certain properties concerning their roots (points where these solutions vanish). The simplest, in a sense paradigmal property, is the following.
Prototheorem. Let
be a nontrivial solution of a sufficiently regular linear ordinary differential equation
of order
. Then
cannot have a root of multiplicity greater or equal than
.
Here by regularity we mean the condition that the operator
has coefficients smooth enough to guarantee that any solution
near any point
in the domain of its definition is uniquely determined by the initial conditions
.
Indeed, if
has a root of multiplicity
, that is, all first
derivatives of
at
vanish, then
by virtue of the equation and hence the by the uniqueness
must be identically zero.
In particular, solutions of first order equation
are nonvanishing, solutions of any second order equation
may have only simple roots etc.
What about quantitative versions?
Theorem (de la Vallee Poussin, 1929). Assume that the coefficients of the LODE
![u^{(n)}+a_1(t)u^{(n-1)}+\cdots+a_{n-1}(t)u'+a_n(t)u=0,\qquad t\in[0,\ell],\qquad (\dag) u^{(n)}+a_1(t)u^{(n-1)}+\cdots+a_{n-1}(t)u'+a_n(t)u=0,\qquad t\in[0,\ell],\qquad (\dag)](https://s0.wp.com/latex.php?latex=u%5E%7B%28n%29%7D%2Ba_1%28t%29u%5E%7B%28n-1%29%7D%2B%5Ccdots%2Ba_%7Bn-1%7D%28t%29u%27%2Ba_n%28t%29u%3D0%2C%5Cqquad+t%5Cin%5B0%2C%5Cell%5D%2C%5Cqquad+%28%5Cdag%29&bg=ffffff&fg=000000&s=0)
are explicitly bounded,
.
Assume that the bounds are small relative to the length of the interval, i.e.,

Then any nontrivial solution of the equation has no more than
isolated roots on
.
Novikov’s counterexample
What about linear systems of the first order?
Consider the system
with
and the norm
explicitly bounded on
. Consider all possible linear combinations
. Can one expect a uniform upper bound for the number of roots of all combinations?
Let
be a polynomial having many zeros on
. Consider the
-system of the form

The first equation defines a nonvanishing function
, the second equation – its derivative which vanishes at all roots of
.
By replacing
by
one can achieve an arbitrarily small sup-norm of the coefficients of this system on the segment
(or even any open complex neighborhood of this real segment). Thus no matter how small are the coefficients, the second component will have the specified number of isolated roots.
Complexification
What about complex valued versions? There is no Rolle theorem for them.
I will describe three possible replacements, Kim’s theorem (1963), nearest in the spirit, and two versions of the argument principle.
Theorem (W. Kim)
Assume that an analytic LODE

is defined in a convex compact subset
of diameter
and the condition (*) holds. Then this equation is disconjugate in
: any solution has at most
isolated roots.
This result follows from the interpolation inequality of the following type: if
is a function holomorphic in
and has
isolated roots there, then
(the maximum modulus norm is assumed).
Consider the equation
on the real interval but with complex-valued coefficients (and solutions). Solutions will be then real parameterized curves
which only exceptionally rarely have roots. Instead of counting roots, one can measure their rotation around the origin
, which is defined as
for any continuous choice of the argument.
Theorem. Assume that

Then rotation of any nontrivial solution
is explicitly bounded:
.
If an analytic LODE with explicitly bounded coefficients is defined, say, on a triangle
, then application of this result to the sides of the triangle yields an explicit upper bound for the number of isolated roots of analytic solutions inside the triangle.
Reference
S. Yakovenko, On functions and curves defined by differential equations, §2.